Though the House group lost one of its eight members — Raúl R. Labrador, R-Idaho, who complained the deal wasn’t tough enough on blocking health benefits for newly legalized immigrants — the agreement could be good news for supporters of an overhaul who have worried about the piecemeal approach advocated by some House GOP leaders.

However, the loss of Labrador from the House group could hurt efforts to woo conservatives.

Indeed, the larger problem has been getting buy-in from the House GOP rank and file for an overhaul that includes a path to citizenship, as the Senate bill would. In order to achieve that goal, Senate Republicans said it’s clear their bill needs to have tougher border security provisions.

“The House side is controlled by Republicans. If you want a bill to pass for the president to sign, it’s got to have Republican support, and right now, the bill that we have in the Senate needs to be strengthened on the border security elements,” Sen. Jeff Flake, R-Ariz., said as he left a briefing with House conservatives in the Republican Study Committee.

“That’s what we’re working on, and we want a bill that passes,” said Flake, who is a member of the bipartisan “gang of eight” that drafted the measure.

Indeed, GOP members of the Senate gang of eight appear to be pushing their fellow group members to back stricter border control measures to pass a bill. For example, Sen. Marco Rubio, R-Fla., suggested on a conservative radio show Tuesday that he might not vote for his own bill if some conservative-backed border amendments don’t pass on the Senate floor.

On Wednesday, Rubio was among the Senate Republicans who traveled across the Capitol to talk to House members. “The general message that I take away from my conversations with Republicans and people across the country is that they are generally prepared to do immigration reform so long as we can ensure that it doesn’t cost the taxpayer money and so long as we can ensure that there isn’t another wave of illegal immigration in the future,” Rubio said.

The bill that Rubio, Flake and six others authored is headed for the Senate floor next week.